If waves only move up and down, why is it that in the ocean you can move forward along a wave? I suppose it's a bit different then a normal wave because you have the action of the crest falling down, but I don't know how that happens either. Where is the forward momentum coming from?
- If you stand on an item too small to experience this mentioned gradient between crest and trough allowing you to glide downhill, you probably are just lifted up vertically and return down. So, no advancement in horizontal direction in comparison to the solid ground / sea floor.
- If the item you are standing on is large enough (yet smaller than the distance to the rise of the next crest), than you advance till the trough. You now overpass the next crest provided the full-filled condition of kinetic energy with you is able to overcompensate the barrier of potential energy then (the next crest of the wave). Some of your kinetic energy is going to be dissipated anyway into (undirected) thermal energy.
Yet, most of the waves of the sea are generated by action of air. Exposing yourself to this external force (i.e. wind) may add / may subtract to your energetic balance. On the other hand, you asked about surfing and this differs from windsurfing.